Question

In: Advanced Math

Let X be a set with infinite cardinality. Define Tx = {U ⊆ X : U...

Let X be a set with infinite cardinality. Define T_x = {U ⊆ X : U = Ø or X \ U is finite}. Prove that T_x is a topology on X. (T_x is called the Cofinite Topology or Finite Complement Topology.)

Expert Solution

be a set with infinite cardinality .

or \ is finite} .

Defination : A collection of subsets of is said to form a tpology on if it satisfies the following three conditions .

If be an finite collection then

If be an infinite collection then

Now we will prove that the collection or \ } is finite forms a topology on .

by defination of .

Also as \ is a finite set .

Suppose be a finite collection of elements of .

are all finite set .

is a finite set as finite union of finite set is finite.

is a finite set .

suppose be an infinite collection of elements of .

are all finite set .

is a finite set as arbitrary intersection of finite set is finite .

is a finite set .

So satisfies all three conditions to be a topology on Hence the collection or \ is finite } is a topology on

Colby Messinger answered 2 years ago

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